So it's good to see that the president has finally started calling some Republicans. He's even invited some to share a meal. And not just moderates, but also those who have been his most vocal opponents -- like John McCain and his buddy Lindsey Graham.

This, of course, is something completely different. After his re-election, the president was asked about his unusually standoffish behavior with Congress. He took offense.

Look at what happened with me and Speaker John Boehner, he said. "When we went out and played golf, we had a great time. But that didn't get a deal done in 2011." And just recently, a senior administration official told me, "There's this myth in Washington that somehow if we all sat down around a table, Republicans would miraculously be more willing to work with us."

So now the White House has decided it's not so silly after all. The president had dinner with 12 GOP senators Wednesday. The big question: Why now?

There are all kinds of ways to answer that, and I'll get to them in a moment. But let me first say this: Give the president some credit. It's the right thing to do, and he should have done it awhile back. He has nothing to lose, and has never had anything to lose by engaging. And if he is interested in some larger budget deal down the road, this is the only way to figure out if it is at all possible.

So now let's get to motive. Having tried and failed to back Republicans into a corner on the forced spending cuts, the White House woke up Wednesday morning to an ABC News/Washington Post poll showing that -- by a 2-1 margin -- the public supports the sequester cuts in general, although it opposes the ones for military spending.

They're also seeing the president's popularity suffer: It has dropped an average of four points since mid-February.

Worse yet, they're seeing that Americans are starting to blame both sides for the Washington standoff. Not long ago, Republicans bore most of the blame. Not anymore.

The administration's strategy hasn't worked. The White House made the assumption that because it succeeded in getting new revenue during the fiscal cliff negotiations, it would be able to do so again. After all, that's what the 2012 election was about: the middle class versus tax cuts for the wealthy. The argument worked and Republicans caved on the fiscal cliff, so they would cave again.

Just the reverse has happened. The GOP cut a deal in January that included tax increases without commensurate spending cuts. No doubt it was a short-term tactical success for the White House, fresh off its impressive re-election. But in the long run, it could turn out to be a strategic mistake. Why? Because Republicans are now more dug in than ever on taxes -- at least without tax reform or serious entitlement cuts.

It's not without irony that the Democratic win on the fiscal cliff narrowed its chances to get a deal on the forced spending cuts. The deal didn't happen. And the public didn't buy the Armageddon scenario. So the once unthinkable cuts are now real, and the public isn't predicting the end of the world.

All of which gets us back to the White House dinner party. Having failed to secure a short-term truce, maybe the White House is finally thinking big picture. The public doesn't like the military cuts; neither do lots of Republicans, including McCain. Maybe that's the entry into a larger discussion: How can we do this without damaging our military?

The pieces are all there: Graham has already told CNN he would be open to increasing revenues if the administration is willing to do serious entitlement reform. Other senior Republicans have said there's a window this summer to get something done. The president is trying to protect the rest of his agenda -- immigration reform, climate change, gun control -- from falling victim to the budget wars.

So even if this is a cynical and calculated dinner hosted by a president in danger of losing political altitude, I'll take it. Even if the president is doing it to prove it won't work, I'll take it. It's not the Last Supper, but it is the first.